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Mirrors > Home > ILE Home > Th. List > seqeq1 | Unicode version |
Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.) |
Ref | Expression |
---|---|
seqeq1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | fveq2 5534 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | 1, 2 | opeq12d 3801 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | freceq2 6418 |
. . . . 5
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5 | 3, 4 | syl 14 |
. . . 4
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6 | fveq2 5534 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | eqid 2189 |
. . . . . 6
![]() ![]() ![]() ![]() | |
8 | mpoeq12 5956 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 6, 7, 8 | sylancl 413 |
. . . . 5
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10 | freceq1 6417 |
. . . . 5
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11 | 9, 10 | syl 14 |
. . . 4
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12 | 5, 11 | eqtrd 2222 |
. . 3
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13 | 12 | rneqd 4874 |
. 2
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14 | df-seqfrec 10477 |
. 2
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15 | df-seqfrec 10477 |
. 2
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16 | 13, 14, 15 | 3eqtr4g 2247 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-cnv 4652 df-dm 4654 df-rn 4655 df-res 4656 df-iota 5196 df-fv 5243 df-oprab 5900 df-mpo 5901 df-recs 6330 df-frec 6416 df-seqfrec 10477 |
This theorem is referenced by: seqeq1d 10482 seq3f1olemqsum 10531 seq3id 10539 seq3z 10542 iserex 11379 summodclem2 11422 summodc 11423 zsumdc 11424 isumsplit 11531 ntrivcvgap 11588 ntrivcvgap0 11589 prodmodclem2 11617 prodmodc 11618 zproddc 11619 fprodntrivap 11624 ege2le3 11711 |
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